Special Values and Transcendence

特殊价值观与超越

• 批准号: 0903838
• 负责人: Matthew Papanikolas
• 金额: $ 15.62万
• 依托单位: Texas A&M Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-01 至 2013-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0903838&HistoricalAwards=false
• 关键词: Special; Values; Transcendence

The principal investigator proposes several projects in arithmetic geometry and transcendental number theory that focus on the deep links between the analytic and arithmetic information coming from values of special functions. In one project, the investigator plans to continue the study of quantities related to Anderson-Drinfeld motives in positive characteristic, including periods, logarithms, Gamma values, zeta values, multiple zeta values, and L-values. Using the Galois theory of difference equations, he will pursue results about the algebraic independence of these quantities both individually and in combinations. In another project, the investigator will consider new problems which associate function field arithmetic invariants to special values of Goss-Hecke L-series. Finally, the investigator will pursue problems relating finite field hypergeometric functions to L-series associated to Siegel modular forms.Number theory is one of the fundamental branches of mathematics, and it serves as the basis for many applications, including cryptography and coding theory. The proposed research considers questions involving values of analytic functions that somewhat remarkably convey fundamental information about fields of algebraic numbers or geometric objects defined over them. Such questions have their genesis in work of Euler and Gauss, and mathematicians continue to endeavor to unravel their mysteries. Several parts of the project lead naturally to problems for graduate and undergraduate research.

主要研究者在算术几何和超越数论中提出了几个项目，重点研究来自特殊函数值的解析信息和算术信息之间的深层联系。在一个项目中，研究者计划继续研究与Anderson-Drinfeld动机相关的正特征量，包括周期、对数、Gamma值、zeta值、多个zeta值和l值。利用差分方程的伽罗瓦理论，他将研究这些量单独和组合的代数独立性的结果。在另一个项目中，研究者将考虑将函数场算术不变量与Goss-Hecke l -级数的特殊值联系起来的新问题。最后，研究者将研究与有限域超几何函数相关的问题，以及与西格尔模形式相关的l -级数。数论是数学的一个基本分支，它是许多应用的基础，包括密码学和编码理论。提出的研究考虑了涉及解析函数值的问题，这些值在某种程度上显著地传达了关于代数数域或定义在它们之上的几何对象的基本信息。这样的问题在欧拉和高斯的著作中有其起源，数学家们一直在努力解开它们的奥秘。这个项目的几个部分自然会给研究生和本科生的研究带来问题。